Modulation/demodulation apparatus using matrix and anti-matrix

ABSTRACT

When a narrow-band digital filter is used for demodulating a signal which has been modulated by using a plenty of sub-carriers, the number of stages is increased, analysis requires a long time, the transmission speed is not increased, and the circuit size is increased. The modulation circuit operation is considered as multiplication.

DETAILED DESCRIPTION OF INVENTION

[0001] [Engineering Field of this Invention]

[0002] This invention use the carrier of which reflection is not so strong, and is applied to the modulation and demodulation apparatus using quadrature magnitude modulation of many number of sub-carrier and transmit the digital data from each other.

[0003] [Conventional Engineering]

[0004] Such method as transmitting the data using many number of sub-carriers which are modulated by quadrature magnitude modulation is applied to QAM of digital cable TV or to DSL of metal twist-pair and so on. These method concentrate on the frequency of each carriers and demodulate the signal by applying digital filter or FFT using impulse response result as coefficient of filter. For this reason it takes comparatively long time to detect the amplitude of a carrier because of looking until it seems as same continuous wave form.

[0005] [Target to Solve by this Invention]

[0006] To detect accurate result from modulated signal, demodulation circuit such as digital filter or FFT concentrate on frequency of each sub-carrier independently and increase the number of wave form to make difficulty of transmission speed. To make transmission speed high, for example, is to increase the number of carriers and reach so high frequency as to decrease transmission distance greatly.

[0007] [Method to Solve these Difficulty by this Invention]

[0008] By this invention, demodulation circuit do not concentrate on frequency of each sub-carrier independently but pay attention the modulated data consisted by the amount of quadrature amplitude modulation of each sub-carriers, and this construction is seemed to be simultaneous linear equation defining amplitude of each sub-carriers as unknown, so can get the result of amplitude of each sub-carrier by solving this simultaneous linear equation. As simultaneous linear equation can be solved in case of the number of unknown equal to the number of each equation, ideally so as to be able to exchange the number of data by equal number of modulated data. This invention is aimed to build the circuit method by taking the simultaneous linear equation and solving in modulation and demodulation circuit. To take amount of quadrature modulation of many sub-carriers is described using matrix mathematics as following.

[0009] This matrix is square and is constructed equal number of line and column which is two times number of sub-carrier frequency and mean sine wave and cosine wave using one frequency. The elements of this matrix called as modulation matrix are value of trigonometric function, and line means number of sampling of which interval is equal to DA converter frequency, and column means sub-carrier which is sine or cosine of a carrier frequency. The product of this modulation matrix and modulation data matrix of one column responding each sub-carrier is named as the modulated data matrix and is converted by DA converter to analog output respectively. As the lines of modulation matrix are arranged according to DA converting number and is the sine or cosine value of carrier frequency, the product of this modulation matrix and one column modulation data matrix means sum of product of the sine or cosine value at every line and modulation data specified to the sub-carrier, and become modulated data for DA converter input of its every interval. These show that quadrature modulation is described as the simultaneous linear equation.

[0010] From the view of demodulation side about this equation, it seams for modulated data to be the received data detected by AD converter, and modulation data as unknown. As there are modulation data as unknown of two times number kinds of carrier frequency, so there are equations of two times number of carrier frequency. Therefore this simultaneous linear equation can be solved. At demodulation side, the sampling frequency of AD converter is adjusted to the sampling frequency of DA converter of modulation side, and similar data as modulated data is got from AD converter, and demodulated data is got as the product of received data matrix from AD converter and inverse matrix of modulation matrix called demodulation matrix. And more, over-sampling is implemented to this concept, the numbers of over-sampling modulation matrix are created, and the numbers of demodulation matrix, which are the inverse matrix of modulation matrix, become as the numbers of over-sampling. And the over-sampling modulation matrix is composed by inserting each line in over-sampling order position, and the over-sampling demodulation matrix is composed by similar method of insertion. When modulation uses this over-sampling modulation matrix and demodulation uses this over-sampling demodulation matrix, then over-sampling number times demodulated data is got. When modulation uses this over-sampling modulation matrix and demodulation uses individual inverse matrices, then the same numbers of over-sampling demodulated data are got.

[0011] In following, these theory is described by using mathematical equation.

[0012] About the modulation matrix which is the base of this theory,

[0013] Line number: i i=1˜2αn

[0014] Column number: j j=1˜2n

[0015] About above definition

[0016] Number of carrier frequency: n

[0017] Number of over-sampling: α

[0018] According to these definition

[0019] Carrier frequency number: p p=0˜(n−1)

[0020] Original sampling order (without over-sampling): q q=0˜(2n−1)

[0021] Order of over-sampling: r r=1˜α

[0022] Kind of wave: s s=1 indicate cosine wave s=2 indicate sine wave

[0023] About relation of these parameter to line number i and column number j

i=αq+r q=0˜(2n−1) r=1˜α therefore i=1˜2αn

j=2p+s p=0(n−1) s=1 or 2 therefore j=1˜2n

[0024] the element of line number i and column number j is Fj(i) and is defined as

F _(j)(i)=F _(2p+s)(αq+r)

[0025] Frequency of frequency number p: fp

[0026] Angle velocity of frequency number p: ωp

[0027] Number of original sampling in one complete wave form: ρ

[0028] interval time of over-sampling: T_(s) $\begin{matrix} {\omega_{p} = {2\pi \quad f_{p}}} \\ {T_{s} = \frac{1}{\rho \times \alpha \times f_{0}}} \end{matrix}$

[0029] and, the angle of sine and cosine in the element of line number i ${\omega_{p} \times T_{s} \times i} = {\frac{2\pi \quad f_{p}}{\rho \times \alpha \times f_{0}} \times \left( {{\alpha \quad q} + r} \right)}$

[0030] Therefore, the element of modulation matrix F_(j)(i) is described following. $\begin{matrix} {{F_{j}(i)} = {{F_{{2p} + s}\left( {{\alpha \quad q} + r} \right)} = {{\cos \left\{ {\frac{2\pi \quad {fp}}{\rho \times \alpha \times f_{0}} \times \left( {{\alpha \quad q} + r} \right)} \right\} \quad {In}\quad {case}\quad {of}\quad s} = 1}}} \\ {\quad {= {{\sin \left\{ {\frac{2\pi \quad {fp}}{\rho \times \alpha \times f_{0}} \times \left( {{\alpha \quad q} + r} \right)} \right\} \quad {In}\quad {case}\quad {of}\quad s} = 2}}} \end{matrix}$

[0031] The size of modulation matrix is 2αn lines and 2n columns. The size of modulation data matrix is 2n lines and one column, and of which element is described as x_(j) because the line number of modulation data matrix is related to the column number of modulation matrix for the reason of relating each modulation data to the sub-carrier of sine and cosine individually. Equation of quadrature modulation is described as the product of the modulation matrix and modulation data matrix. The size of modulated data matrix, which is the product of the modulation matrix and modulation data matrix, is 2αn lines and one column, and the element of modulated data matrix is described as d_(i) according to line number of modulation matrix. As the element of modulated data matrix d_(i) is the amount of quadrature modulation of each sub-carrier at every sampling time, the equation of modulation is described by matrix as following

(F _(j)(i))×(x _(j))=(d _(i))

[0032] and this equation is described by elements as following ${\begin{pmatrix} {{F_{1}(1)},{F_{2}(1)},{F_{3}(1)},{\cdots \quad {F_{{2n} - 1}(1)}},{F_{2n}(1)},} \\ {{F_{1}(2)},{F_{2}(2)},{F_{3}(2)},{\cdots \quad {F_{{2n} - 1}(2)}},{F_{2n}(2)},} \\ {{F_{1}(3)},{F_{2}(3)},{F_{3}(3)},{\cdots \quad {F_{{2n} - 1}(3)}},{F_{2n}(3)},} \\ {\quad \vdots} \\ {{F_{1}\left( {2\alpha \quad n} \right)},{F_{2}\left( {2\alpha \quad n} \right)},{F_{3}\left( {2\alpha \quad n} \right)},{\cdots \quad {F_{{2n} - 1}\left( {2\alpha \quad n} \right)}},{F_{2n}\left( {2\alpha \quad n} \right)},} \end{pmatrix} \times \begin{pmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{j} \\ \vdots \\ x_{2n} \end{pmatrix}} = \begin{pmatrix} d_{1} \\ d_{2} \\ \vdots \\ d_{i} \\ \vdots \\ d_{2\alpha \quad n} \end{pmatrix}$

[0033] And this equation is described by simultaneous linear equation is as following

F ₁(1)x ₁ +F ₂(1)x ₂ +F ₃(1)x ₃ + . . . +F _(2n−1)(1)x _(2n−1) +F _(2n)(1)x _(2n) =d ₁

F ₁(2)x ₁ +F ₂(2)x ₂ +F ₃(2)x ₃ + . . . +F _(2n−1)(2)x _(2n−1) +F _(2n)(2)x _(2n) =d ₂

F ₁(3)x ₁ +F ₂(3)x ₂ +F ₃(3)x ₃ + . . . +F _(2n−1)(3)x _(2n−1) +F _(2n)(3)x _(2n) =d ₂

F ₁(2αn)x ₁ +F ₂(2αn)x ₂ +F ₃(2αn)x ₃ + . . . +F _(2n−1)(2αn)x _(2n−1) +F _(2n)(2αn)x _(2n) =d _(2αn)

[0034] Depend on this simultaneous linear equation, the product of initially determined F_(j)(i) and modulation data x_(j) related by j, summed together through all j resulting to d_(j) as the input data of DA converter at sampling number i, and is converted to analog output. d_(j) can be got by multiplying and accumulating in every sampling interval. Following is described about the treatment in receiving side. The matrix of which elements F_(r0,j)(g) is picked up from modulation matrix by implementing the over-sampling order number r=r₀, then is represented as following. $\begin{matrix} \begin{matrix} {{F_{{r0},{{2p} + s}}(\quad q)} = {{\cos \left\{ {\frac{2\pi \quad f_{p}}{\rho \times \alpha \times f_{0}} \times \left( {{\alpha \quad q} + r_{0}} \right)} \right\} \quad {in}\quad {case}\quad {of}\quad s} = 1}} \\ {\quad {{\sin \left\{ {\frac{2\pi \quad f_{p}}{\rho \times \alpha \times f_{0}} \times \left( {{\alpha \quad q} + r_{0}} \right)} \right\} \quad {in}\quad {case}\quad {of}\quad s} = 2}} \end{matrix} \\ {q = {0 \sim \left( {{2n} - 1} \right)}} \end{matrix}$

[0035] The first line of this matrix is first r₀ line of modulation matrix, and the other line is picked up from modulation matrix every α line from r₀ to constructing 2n lines. The first line of related modulated data matrix is first r₀ line of modulated data matrix and the other line is picked up from modulated data matrix every α line from r₀ to constructing 2n lines, and of which element is described as d_(r0,q) as following.

d _(r0,q) =d _((αq+r) ₀ ₎

[0036] The modulation equation of above matrix picking up by sampling order r₀ is described by matrix as following

(F _(r0,2p+s)(q))×(x _(2p+s))=(d _(r0,) _(q) )

[0037] and this equation is described by elements as following ${\begin{pmatrix} {{F_{{r\quad 0},1}(0)},{F_{{r\quad 0},2}(0)},{F_{{r\quad 0},3}(0)},{\cdots \quad {F_{{r\quad 0},{{2n} - 1}}(0)}},{F_{{r\quad 0},{2n}}(0)}} \\ {{F_{{r\quad 0},1}(1)},{F_{{r\quad 0},2}(1)},{F_{{r\quad 0},3}(1)},{\cdots \quad {F_{{r\quad 0},{{2n} - 1}}(1)}},{F_{{r\quad 0},{2n}}(1)}} \\ {\quad \vdots} \\ \begin{matrix} {{{F_{{r\quad 0},1}\left( {{2\quad n} - 1} \right)},{F_{{r\quad 0},2}\left( {{2\quad n} - 1} \right)},{F_{{r\quad 0},3}\left( {{2\quad n} - 1} \right)},\cdots}\quad} \\ {\quad {{F_{{r\quad 0},{{2n} - 1}}\left( {{2\quad n} - 1} \right)},{F_{{r\quad 0},{2n}}\left( {{2\quad n} - 1} \right)},}} \end{matrix} \end{pmatrix} \times \begin{pmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{j} \\ \vdots \\ x_{2n} \end{pmatrix}} = \begin{pmatrix} d_{{r\quad 0},0} \\ d_{{r\quad 0},1} \\ \vdots \\ \vdots \\ \vdots \\ d_{{r\quad 0},{2\alpha \quad n}} \end{pmatrix}$

[0038] And this equation is described by simultaneous linear equation as following

F _(r0,1)(0)x ₁ +F _(r0,2)(0)x ₂ +F _(r0,3)(0)x ₃ + . . . F _(r0,2n−1)(0)x _(2n−1) +F _(r0,) _(2n)(0)x _(2n) =d _(r0,0)

F _(r0,1)(1)x ₁ +F _(r0,2)(1)x ₂ +F _(r0,3)(1)x ₃ + . . . F _(r0,2n−1)(1)x _(2n−1) +F _(r0,) _(2n)(1)x _(2n) =d _(r0,1)

F _(r0,1)(2n−1)x ₁ +F _(r0,2)(2n−1)x₂ +F _(r0,3)(2n−1)₃ + . . F _(r0,2n−1)(2n−1 )x _(2n−1) +F _(r0,2n)(2n−1)x _(2n) =d _(r0,2n−1)

[0039] In this simultaneous linear equation, d_(r0,0)˜d_(r0,2n−1) are similarly got as receiving data by AD converter. For the demodulation side to detect the modulation data x₁˜x_(2n), the inverse matrix of the modulation matrix of which element is (F_(r0,2p+s)) is applied to solving this simultaneous linear equation. The element of inverse matrix of modulation matrix is described as G_(r0,j)(q) ${\begin{pmatrix} {{G_{{r\quad 0},1}(0)},{G_{{r\quad 0},2}(0)},{G_{{r\quad 0},3}(0)},{\cdots \quad {G_{{r\quad 0},{{2n} - 1}}(0)}},{G_{{r\quad 0},{2n}}(0)}} \\ {{G_{{r\quad 0},1}(1)},{G_{{r\quad 0},2}(1)},{G_{{r\quad 0},3}(1)},{\cdots \quad {G_{{r\quad 0},{{2n} - 1}}(1)}},{G_{{r\quad 0},{2n}}(1)}} \\ {\quad \vdots} \\ {{{G_{{r\quad 0},1}\left( {{2n} - 1} \right)},{G_{{r\quad 0},2}\left( {{2n} - 1} \right)},{G_{{r\quad 0},3}\left( {{2n} - 1} \right)},\cdots}\quad} \\ {\quad {{G_{{r\quad 0},{{2n} - 1}}\left( {{2n} - 1} \right)},{G_{{r\quad 0},{2n}}\left( {{2n} - 1} \right)},}} \end{pmatrix} \times \begin{pmatrix} d_{{r\quad 0},0} \\ d_{{r\quad 0},1} \\ \vdots \\ d_{{r\quad 0},{{2\quad n} - 1}} \end{pmatrix}} = \begin{pmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{2n} \end{pmatrix}$

[0040] And this equation is described by simultaneous linear equation as following

G _(r0,1)(0)d _(r0,0) +G _(r0,2)(0)d _(r0,1) +G _(r0,3)(0)d _(r0,2) + . . . G _(r0,2n−1)(0)d _(r0,2n−2) +G _(r0,2n)(0)d _(r0,2n−1) =x ₁

G _(r0,1)(1)d _(r0,0) +G _(r0,2)(1)d _(r0,1) +G _(r0,3)(1)d _(r0,2) + . . . G _(r0,2n−1)(1)d _(r0,2n−2) +G _(r0,2n)(0)d _(r0,2n−1) =x ₂

G _(r0,1)(2n− 1) d _(r0,0) +G _(r0,2)(2n−1)d _(r0,1) +G _(r0,3)(2n−1)d _(r0,2) + . . G _(r0,2n−1)(2n−1 )d _(r0,2n−2) +G _(r0,2n)(2n−1)d _(r0,2n−1) =x _(2n)

[0041] According to this simultaneous linear equation, the inverse matrix of the matrix which is picked up the same over-sampling number line from the over-sampling modulation matrix, and the receiving data coming from AD converter at every r₀ sampling interval, are multiplied and summed together by two times number of carrier frequency accumulators, continuously until the end of one frame of modulation, and then the demodulated data of all sub-carriers are got.

[0042] Mathematically, the size of receiving data matrix is one column matrix, and about the construction of inverse matrix, the line number is relating to column number of modulation matrix, and the column number is relating to line number of modulation matrix and it look like exchange the line and column.

[0043] As the number of over-sampling is α, the number of α inverse matrix are created from modulation matrix and are got the number of α kind of demodulated data by this operation. When demodulation starts from the first line of demodulation matrix synchronizing to the receiving data of the first line of modulation matrix, α kind of demodulated data are equal one after another because of only one kind of modulation data. When demodulation starts from several sampling later than the first line of demodulation matrix but not over a sampling, the number of same demodulation data decrease according to the number of several sampling delay. When demodulation starts from over a sampling after the first line of demodulation matrix, no same demodulation data is got because the one frame time belonging to the one operation of demodulation matrix spread two frame time belonging to first modulated data matrix and of next modulated data matrix and the receiving data are constructed by first modulation data and next modulation data. This property is applied to synchronization of modulation and demodulation. The matrix, of which column is picked up from one column of demodulation matrix and is constructed other column by shifting one over-sampling interval from each other to the end of the line number of demodulation matrix, is created, and demodulation operation is applied to any receiving data from AD converter by this shifted matrix, and synchronize point is found by the columns number of same demodulated data.

[0044] The meaning of demodulation, which use the matrix composed the lines from over-sampling number of inverse matrix placed at over-sampling proper timing under the condition of synchronizing with modulation, is that there are over-sampling number of simultaneous linear equation, and over-sampling number of same modulated data are solved. The products of over-sampling demodulation matrix and receiving data matrix from AD converter are summed together, and over-sampling number times of similar modulated data are got, and this contribute the reduction of electrical circuits of multiplier and accumulator.

[0045] Next explaining the adjusting method in following

[0046] The distortion, which is created by the parameter of line such as twist-pair between terminals, which is created by sampling timing difference between DA converter and AD converter, should be adjusted to get the correct demodulated data. Before the practical communication under the circumstance of decided parameter of line or DA or AD converter, test communication is done to get the parameter of adjustment.

[0047] Modulation data of sub-carrier frequency number p are x_(2p+1) for cosine and x_(2p+2) for sine and the phase of these wave is shifted θ_(p) in the receiving data by the parameter of communication line or sampling timing difference of DA and AD converter.

[0048] The phase shifted form of the wave described as following.

x _(2p+1) cos(ω_(p) t+θ _(p))=x _(2p+1) cos θ_(p) cos ω_(p) t−x _(2p+1) sin θ_(p) sin ω_(p) t

x _(2p+2) sin(ω_(p) t+θ _(p))=x _(2p+2) cos θ_(p) sin ω_(p) t−x _(2p+2) sin θ_(p) cos ω_(p) t

[0049] In demodulation side, the amount of these wave is got as the receiving data, and practical demodulated data β_(2p+1), for cosine and β_(2p+2) for sine, which is demodulated by the operation of receiving data and demodulation matrix about cosine and sine independently, are got as coefficient of cos ω_(p)t and sin ω_(p)t. And practical demodulated data is described mathematically as following.

β_(2p+1) =x _(2p+1) cos θ_(p) +x _(2p+2) sin θ_(p)

β_(2p+2) =−x _(2p+1) sin θ_(p) +x _(2p+2) cos θ_(p)

[0050] Practical demodulated data of each sampling index r are described as β_(r,2p+1) and β_(r,2p+2), which is detected by the operation of partial demodulation matrix and partial receiving data matrix of each sampling index. This demodulated data are described by use of raff equal symbol because of being distorted by noise and phase shift. And equation is described as following.

β _(r,2p+1) ≈x _(2p+1) cos θ_(p) +x _(2p+2) sin θ_(p)

βr,2p+2 ≅−x _(2p+1) sin θ_(p) +x _(2p+2) cos θ_(p)

[0051] Difference is took in spite of raff equal symbol, and amount of square of these difference is described as δ_(p) ² and is differentiated by θ_(p) to apply minimum square method. $\begin{matrix} {{\frac{\partial\quad}{\partial\theta_{p}}\delta_{p}^{2}} = {{2\left( {{x_{{2p} + 1}\sin \quad \theta_{p}} - {x_{{2p} + 2}\cos \quad \theta_{p}}} \right){\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2p} + 1}}}} +}} \\ {{2\left( {{x_{{2p} + 1}\cos \quad \theta_{p}} + {x_{{2p} + 2}\sin \quad \theta_{p}}} \right){\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2p} + 2}}}}} \end{matrix}$

[0052] The modulation data of the test communication before practical communication is described following

x _(2p+1) =x _(2p+2) =x _(test)≢0

[0053] To get θ_(p) by applying minimum square method ${\frac{\partial\quad}{\partial\theta_{p}}\delta_{p}^{2}} = 0$

[0054] and get the result for θ_(p) in following ${\tan \quad \theta_{p}} = \frac{{\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2\quad p} + 1}}} - {\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2\quad p} + 2}}}}{{\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2\quad p} + 1}}} + {\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2\quad p} + 2}}}}$

[0055] And cos θ_(p) or sinθ_(p) is calculated by tanθ_(p).

[0056] As modulation data is determined as the mean value of over-sampling number of practical demodulation data, so is described following $\begin{matrix} {x_{{2p} + 1} \cong {{\cos \quad \theta_{p} \times \left\{ {\frac{1}{\alpha}{\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2p} + 1}}}} \right\}} - {\sin \quad \theta_{p}\left\{ {\frac{1}{\alpha}{\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2p} + 2}}}} \right\}}}} \\ {x_{{2p} + 2} \cong {{\sin \quad \theta_{p} \times \left\{ {\frac{1}{\alpha}{\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2p} + 1}}}} \right\}} + {\cos \quad \theta_{p}\left\{ {\frac{1}{\alpha}{\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2p} + 2}}}} \right\}}}} \end{matrix}$

[0057] The modulation data x_(test) before the practical communication is already known at receiving side and described as following.

[0058] In the column number 2p+1

x _(test)≅cos θ_(p) ×{overscore (D)} _(2p+1)(test)−sin θ_(p) ×{overscore (D)} _(2p+2)(test)

[0059] In the column number 2p+2

x _(test)≅sin θ_(p) ×{overscore (D)} _(2p+1)(test)+cos θ_(p) ×{overscore (D)} _(2p+2)(test)

[0060] In these equation $\begin{matrix} {{\overset{\_}{D}}_{{2p} + 1} = {\frac{1}{\alpha}{\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2p} + 1}}}}} \\ {{\overset{\_}{D}}_{{2p} + 2} = {\frac{1}{\alpha}{\sum\limits_{r = 1}^{\alpha}\quad \beta_{r,{{2p} + 2}}}}} \end{matrix}$

[0061] {overscore (D)} followed by (test) mean the practical demodulated data in test communication and mean value of a demodulated data.

[0062] The demodulated data differ from the modulation data in demodulation side, and is adjusted by the way that ratio of amplitude of test communication and practical communication is equal, and adjustment equation is as following. $\begin{matrix} {x_{{2p} + 1}->\frac{x_{test} \times \left( {{\cos \quad \theta_{p} \times {\overset{\_}{D}}_{{2p} + 1}} - {\sin \quad \theta_{p}{\overset{\_}{D}}_{{2p} + 2}}} \right)}{{\cos \quad \theta_{p} \times {{\overset{\_}{D}}_{{2p} + 1}({test})}} - {\sin \quad \theta_{p} \times {{\overset{\_}{D}}_{{2p} + 2}({test})}}}} \\ {x_{{2p} + 2}->\frac{x_{test} \times \left( {{\sin \quad \theta_{p} \times {\overset{\_}{D}}_{{2p} + 1}} + {\cos \quad \theta_{p}{\overset{\_}{D}}_{{2p} + 2}}} \right)}{{\sin \quad \theta_{p} \times {{\overset{\_}{D}}_{{2p} + 1}({test})}} + {\cos \quad \theta_{p} \times {{\overset{\_}{D}}_{{2p} + 2}({test})}}}} \end{matrix}$

[0063] Therefore demodulation data is adjusted from such influence of communication line.

[0064]FIG. 1 is block diagram of total system of modulation and demodulation. In modulation side, the data of modulation matrix ROM, of which address is specified by the address counter for modulation, and the modulation data are multiplied responding to cosine and sine of all sub-carrier frequency individually and are added all product to the data of DA converter.

[0065] In demodulation side, analog signal is converted by the AD converter to digital signal and is multiplied with the data of demodulation matrix ROM1, of which address is specified by the address counter for demodulation, about cosine and sine of all sub-carrier frequency individually in every sampling interval and are accumulated until the end of one modulation block, and adjusted every end of block about phase shift to the adjusted demodulated data.

[0066] About the synchronization between the address counter for modulation and address counter for demodulation, ROM named as ROM2 is picked up the memory data belonging to one carrier frequency from the demodulation ROM1, and the data of the other block of memory are moved some address each other to the end of memory address. The receiving data from AD converter is multiplied and accumulated with the memory data of cosine and sine individually in numbers of shift, and at the end of one modulation block, is adjusted about the phase shift and sent to synchronization circuit. In the synchronization circuit, the demodulated data is arranged according to the order of address shift, and the first data of same data series nearly as a is found out, and reset the address counter for demodulation adjusting the delay between the address counter and the number of shift.

[0067]FIG. 2 is block diagram of modulation. The maximum number of address counter for modulation is 2αn. Address of modulation ROM is 2αn wide and number of data buss is 2nW(word) wide. 1W of modulation ROM store Fj(i). This ROM send the data of 2nW wide responding i, which is specified as the index of sampling, to the modulation data of 2n. In every clock, the product of each modulation data and specified Fj(i) of 1W wide are summed together for all number of modulation data and is converted by DA converter to analog to the communication line. Before the practical communication, the test communication is made by the modulation data x_(test)

[0068]FIG. 3 is the block diagram of demodulation to get the demodulated data before adjustment of phase. Maximum address of address counter for demodulation is 2αn. Address of demodulation ROM1 is 2αn wide and data of it is 2nW wide (number of 2n of 1W wide ROM). The data of 1W of demodulation ROM1 is specified Gj(i). The analog signal from communication line is converted to digital by AD converter at same sampling interval as the clock of DA converter in modulation side.

[0069] 2nW wide data is read out from demodulation ROM1 at every clock, and every 1W related to cosine or sine of sub-carrier frequency individually is multiplied with the receiving data at this moment, and is accumulated individually until the number of 2αn, and is divided by α as the demodulated data {overscore (D)}_(2p+1), {overscore (D)}_(2p+2) before adjustment of phase.

[0070] Using the mean value of demodulated data of cosine and sine of same carrier of number p as {overscore (D)}_(2p+1), {overscore (D)}_(2p+2) drawn in FIG. 3 block diagram, FIG. 4 shows the adjustment circuit diagram of phase and magnitude.

[0071] About the basic circuit operation in FIG. 4, when the system is reset or is happen to change the parameter of adjustment according the condition of communication line, by the modulation data x_(test) which is determined to exchange at initial test communication by both modulation side and demodulation side, the parameter of adjustment is set. And after this operation, practical communication start and use this parameter for adjustment calculation.

[0072] The mean value of demodulation data {overscore (D)}_(2p+1) and {overscore (D)}_(2p+2) are squared by block of multiplyer1 and multiplyer2 and are added each other by adder2 and are stored by DFF1 at the end of initial test communication after system reset. And this data is sent to DFF1, DFF2 and DFF3 at only one time after initial test communication by one clock in the transmission unit frame time as 2n α as numbers conversions of DA and AD, and stored until next reset after first one.

[0073] And by the similar operation, the added value of {overscore (D)}_(2p+1) and {overscore (D)}_(2p+2) is stored in DFF2 and difference value of {overscore (D)}_(2p+1) and {overscore (D)}_(2p+2) is stored in DFF3. These three stored data and x_(test) of initial test data of communication are stored as the parameter of adjustment until next system reset.

[0074] As the parameter of adjustment for multiplier.7 and multiplier.8 are as following.

x_(test)

[0075] data stored in DFF1 {overscore (D)} _(2p+1) ²(test)+{overscore (D)} _(2p+2) ²(test)

data stored in DFF2 {overscore (D)} _(2p+1)(test)+{overscore (D)}_(2p+2)(test)

data stored in DFF3 {overscore (D)} _(2p+1) (test)−{overscore (D)} _(2p+2)(test)

[0076] In practical communication after the initial test communication, the mean value of demodulated data in every one frame {overscore (D)}_(2p+1) is stored in DFF4 and {overscore (D)}_(2p+2) is stored in DFF5 and is renewed at the interval of 2n α number of clock.

[0077] Stored data in some operation blocks is as following.

multiplier.3 ({overscore (D)} _(2p+1)(test)+{overscore (D)} _(2p+2)(test))×{overscore (D)}_(2p+1)

multiplier.4 ({overscore (D)} _(2p+1)(test)+{overscore (D)}_(2p+2)(test))×{overscore (D)} _(2p+2)

multiplier.5 ({overscore (D)} _(2p+1)(test)−{overscore (D)} _(2p+2)(test))×{overscore (D)} _(2p+1)

multiplier.6 ({overscore (D)} _(2p+1)(test)−{overscore (D)} _(2p+2) (test))×{overscore (D)} _(2p+2)

[0078] and

difference.2 ({overscore (D)} _(2p+1)(test)+{overscore (D)} _(2p+2) (test))×{overscore (D)} _(2p+1)−({overscore (D)} _(2p+1)(test)−{overscore (D)} _(2p+2) (test))×{overscore (D)} _(2p+2)

adder.3 ({overscore (D)} _(2p+1)(test)+{overscore (D)} _(2p+2)(test))×{overscore (D)} _(2p+2)+({overscore (D)} _(2p+1)(test)−{overscore (D)} _(2p+2)(test))×{overscore (D)} _(2p+1)

[0079] and modulation data in test communication x_(test) is took in place $\begin{matrix} {x_{test} \times \begin{Bmatrix} {{\left( {{{\overset{\_}{D}}_{{2p} + 1}({test})} + {{\overset{\_}{D}}_{{2p} + 2}({test})}} \right) \times {\overset{\_}{D}}_{{2p} + 1}} -} \\ {\left( {{{\overset{\_}{D}}_{{2p} + 1}({test})} - {{\overset{\_}{D}}_{{2p} + 2}({test})}} \right) \times {\overset{\_}{D}}_{{2p} + 2}} \end{Bmatrix}} & {{multiplier}.\quad 7} \\ {x_{test} \times \begin{Bmatrix} {{\left( {{{\overset{\_}{D}}_{{2p} + 1}({test})} + {{\overset{\_}{D}}_{{2p} + 2}({test})}} \right) \times {\overset{\_}{D}}_{{2p} + 2}} +} \\ {\left( {{{\overset{\_}{D}}_{{2p} + 1}({test})} - {{\overset{\_}{D}}_{{2p} + 2}({test})}} \right) \times {\overset{\_}{D}}_{{2p} + 1}} \end{Bmatrix}} & {{multiplier}.\quad 8} \end{matrix}$

[0080] finally, the amount of squared mean value of demodulated data is took in place $\begin{matrix} {\frac{x_{test} \times \left\{ {{\left( {{{\overset{\_}{D}}_{{2p} + 1}({test})} + {{\overset{\_}{D}}_{{2p} + 2}({test})}} \right) \times {\overset{\_}{D}}_{{2p} + 1}} - {\left( {{{\overset{\_}{D}}_{{2p} + 1}({test})} - {{\overset{\_}{D}}_{{2p} + 2}({test})}} \right) \times {\overset{\_}{D}}_{{2p} + 2}}} \right\}}{{{\overset{\_}{D}}_{{2p} + 1}^{2}({test})} + {{\overset{\_}{D}}_{{2p} + 2}^{2}({test})}}->x_{{2p} + 1}} & {{divider}.\quad 1} \\ {\frac{x_{test} \times \left\{ {{\left( {{{\overset{\_}{D}}_{{2p} + 1}({test})} + {{\overset{\_}{D}}_{{2p} + 2}({test})}} \right) \times {\overset{\_}{D}}_{{2p} + 2}} - {\left( {{{\overset{\_}{D}}_{{2p} + 1}({test})} - {{\overset{\_}{D}}_{{2p} + 2}({test})}} \right) \times {\overset{\_}{D}}_{{2p} + 1}}} \right\}}{{{\overset{\_}{D}}_{{2p} + 1}^{2}({test})} + {{\overset{\_}{D}}_{{2p} + 2}^{2}({test})}}->x_{{2p} + 2}} & {{divider}.\quad 2} \end{matrix}$

[0081] Therefore, the demodulated data is adjusted.

[0082] Demodulation data is got at every 2αn number of data from AD converter which is the frame of one modulation. And phase adjustment of {overscore (D)}_(2p+1), {overscore (D)}_(2p+2) in every carrier may be done at 2αn clock interval. To increase the efficiency of usage of circuit, the amount of circuit decrease by using a time sharing phase adjustment circuit.

[0083] The time sharing phase adjustment circuit is represented in FIG. 5. {overscore (D)}_(2p+1) and {overscore (D)}_(2p+2) detect and store demodulated data before phase adjustment, and sent one block by selector o to phase adjustment circuit at every one frame. In time sharing phase adjustment circuit, number of n registers in spite of DFF1, DFF2 and DFF3 which are represented in phase adjustment circuit FIG. 4 for one sub-carrier frequency, are selected one by one by selector 1 synchronized to selector 0, and are stored as the parameter of every sub-carrier in one round of selector 1 when the parameter is decided in system.

[0084] In practical communication, the parameters responding to the index of sub-carrier are selected by the selector 2 synchronizing to the selector 0 in spite of operation DFF4 and DFF5. The calculation in the circuit is done ideally by the pip-line operation. Therefore phase adjusted demodulation data are sent from the time sharing phase adjustment circuit continuously.

[0085] Demodulation circuit block diagram for synchronization is represented in FIG. 6. Address counter for demodulation is used for this circuit and output 2αn addresses. Demodulation ROM2 has 2αn addresses and 4nW wide data bus. ROM2 pick up the memory data G_(2p+1)(i) and G_(2p+2)(i) belonging to one carrier frequency p from the demodulation ROM1 and the data of the other block of memory are moved a address each other to the end of memory address. Data of this ROM2 is 4nW wide which is 2nW numbers of cosine data and 2nW number of sine data. In one clock interval, demodulation ROM2 output 4nW wide data, with which the data from AD converter is multiplied and accumulated by a number individually and selected as {overscore (D)}_(2p+1) and {overscore (D)}_(2p+2) to the time sharing phase adjustment circuit. Time sharing phase adjustment circuit is operated not by the half clock but by two times clock of address counter. At least, one out put of the time sharing adjustment circuit is sent to synchronization circuit represented in FIG. 7 to detect frame synchronized signal. Serise of adjusted demodulated data for synchronization are shifted by equal or less than α number of DFF, and each shifted data are compared, and synchronization signal is output in case of all equal data. This synchronization signal is shifted as long as the delay between synchronization circuit and address counter for demodulation, and reset the counter for demodulation to synchronize the counter for demodulation with the counter for modulation of the other side terminal.

[0086] About method of sub-carrier frequency determination described in claim 2, the cosine and sine wave equation of sub-carrier number p are $\begin{matrix} {\cos \left\{ {\frac{2\pi \quad f_{\rho}}{\rho \times \alpha \times f_{0}}\left( {{\alpha \quad q} + r} \right)} \right\}} \\ {\sin \left\{ {\frac{2\pi \quad f_{\rho}}{\rho \times \alpha \times f_{0}}\left( {{\alpha \quad q} + r} \right)} \right\}} \end{matrix}$

[0087] and following equation for sub-carrier frequency reference ${\sum\limits_{q = 0}^{({{2n} - 1})}\quad {\cos^{2}\left\{ {\frac{2\pi \quad f_{\rho}}{\rho \times \alpha \times f_{0}}\left( {{\alpha \quad q} + r} \right)} \right\}}} = {\sum\limits_{q = 0}^{({{2n} - 1})}\quad {\sin^{2}\left\{ {\frac{2\pi \quad f_{\rho}}{\rho \times \alpha \times f_{0}}\left( {{\alpha \quad q} + r} \right)} \right\}}}$

[0088] and this equation is solved by many ƒ_(p) by which is made matrix and inverse matrix. And the difference of range in the inverse matrix elements are not so wide or so near to zero that the proper ƒ_(p) are selected.

[0089] The transmission speed is as following

[0090] The sampling clock of DA converter is CLK and most high frequency of sub-carrier is ƒ₀ and ρ is sampling numbers in one wave of most high frequency of sub-carrier

CLK=ρ×α×ƒ₀=ραƒ₀

[0091] The number of sampling of one frame in modulation is same as the number of address of modulation ROM, that is 2αn. The bits wide of modulation data are specified as A, then the total bits wide of modulation data are 2nA. Therefore the transmission speed is following. ${\frac{CLK}{{2\alpha \quad n}\quad} \times 2\quad {nA}} = {{CLK} \times \frac{A}{\alpha}}$

[0092] Another method of synchronization block diagram is represented in FIG. 8. Although the modulation data are same in first two round address for modulation and same in second two round address of modulation, the modulation data of the sub-carrier which is specified as the use of synchronization should be different in first two round address from in second two round address. The data of demodulation ROM1 which is used for synchronization is called as synchronization ROM. The address of synchronization ROM is connected to the continuous address counter.

[0093] The product of the data from AD converter and the data of synchronization ROM is accumulated for the one round address of demodulation circuit, and the accumulator, which start accumulation from every address for one round addresses to output result continuously, is installed in synchronization circuit. These value of the accumulator after multiplying are not different from each other for the term of same modulation data, but are different from each other for the term of the different modulation data. In synchronization circuit, this property contribute to make synchronization signal which is output by the comparator indication the equality or the difference between two series adjusted demodulated data. Multiplier and accumulator starting from every address is represented in FIG. 9.

[0094] The data from AD converter and the data of synchronization ROM are multiplied by the circuit of multiplier, and send to the circuit of accumulator. The output of accumulator is sent to DFF6 by every clock, and is returned to accumulator to be added with next data one after another.

[0095] The carry-out signal, which is output at every one round of address counter, reset DFF6, and another DFF7 store the last accumulated value as the data of accumulation. This accumulating operation is same as in demodulation circuit.

[0096] Until next accumulation from previous this one, the accumulator, which start accumulation from every address for one round addresses, is operated as following. The accumulated data until this time of previous round in DFF8 is subtracted from the previous accumulated data in DFF7 and is added by newly accumulated data until this time in DFF6, and putout this data at every address.

[0097] To achieve this operation, dual-port RAM, of which highest write address bit is connected through TFF to carry-out of address counter and the lower address is connected to address counter output, and of which read address is similar to write address with inverted highest address, output the previous round data, which are accumulated and stored in DFF8, and above subscription is got.

[0098] Watching whether the number of address counter for demodulation is continuous or not at synchronization, the demodulation data is adjusted in case of over-ride by subscripting the product and incase of less number adding the product.

EXAMPLE

[0099] Number of carrier frequency n=4

[0100] Number of over-sampling α=4

[0101] Kind of wave s s=1 indicate cosine wave s=2 indicate sine wave

[0102] about modulation matrix

[0103] number of line i i=1˜32

[0104] number of column j j=1˜8

[0105] sub-carrier frequency number p p=0˜3

[0106] Original sampling order q q=0˜7

[0107] Order of over-sampling r r=1˜4

i=αq+r=4q+r

j=2p+s

[0108] the elements of matrix

F _(i)(i)=F _(2p+s)(4q+r)

[0109] Number of original sampling in one complete wave form ρ=2.399 $\begin{matrix} {{F_{{2p} + s}\left( {{4q} + r} \right)} = {\cos \left\{ {\frac{2\pi \quad f_{p}}{9.596\quad f_{0}}\left( {{4q} + r} \right)} \right\}}} & {{{{in}\quad {case}\quad {of}\quad s} = 1}} \\ {{\sin \left\{ {\frac{2\pi \quad f_{p}}{9.596\quad f_{0}}\left( {{4q} + r} \right)} \right\}}} & {{{{in}\quad {case}\quad {of}\quad s} = 2}} \end{matrix}$

[0110] f₀=1.0423 MHz

[0111] f₁=0.7809 MHz

[0112] f₂=0.6255 MHz

[0113] f₃=0.4684 MHz

[0114] basic sampling interval 383.7 nSec

[0115] over-sampling interval 95.9 nSec demodulation ROM2(conbined ROM for synchronization) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q r i 1 2 3 4 5 6 7 8 0 1 1 9D62 8CD2 8CD2 AC89 AC89 9C5E 9C5E 7AA0 2 2 97C8 97FB 97FB 9FE6 9FE6 AA6B AA6B 6DA9 3 3 8C8B 9D6B 9D6B 8E22 8E22 B0DB B0DB 650B 4 4 8051 9AE3 9AE3 7B6D 7B6D AE2A AE2A 6216 1 1 5 8CD2 AC89 AC89 9C5E 9C5E 7AA0 7AA0 AF18 2 6 97FB 9FE6 9FE6 AA6B AA6B 6DA9 6DA9 A655 3 7 9D6B 8E22 8E22 B0DB B0DB 650B 650B 9945 4 8 9AE3 7B6D 7B6D AE2A AE2A 6216 6216 89E8 2 1 9 AC89 9C5E 9C5E 7AA0 7AA0 AF18 AF18 7BD9 2 10 9FE6 AA6B AA6B 6DA9 6DA9 A655 A655 7DB4 3 11 8E22 B0DB B0DB 650B 650B 9945 9945 80A2 4 12 7B6D AE2A AE2A 6216 6216 89E8 89E8 8461 3 1 13 9C5E 7AA0 7AA0 AF18 AF18 7BD9 7BD9 85E0 2 14 AA6B 6DA9 6DA9 A655 A655 7DB4 7DB4 81E8 3 15 B0DB 650B 650B 9945 9945 80A2 80A2 7EA6 4 16 AE2A 6216 6216 89E8 89E8 8461 8461 7C62 4 1 17 7AA0 AF18 AF18 7BD9 7BD9 85E0 85E0 9D62 2 18 6DA9 A655 A655 7DB4 7DB4 81E8 81E8 97C8 3 19 650B 9945 9945 80A2 80A2 7EA6 7EA6 8C8B 4 20 6216 89E8 89E8 8461 8461 7C62 3E98 EE06 5 1 21 AF18 7BD9 7BD9 85E0 85E0 9D62 9D62 8CD2 2 22 A655 7DB4 7DB4 81E8 81E8 97C8 97C8 97FB 3 23 9945 80A2 80A2 7EA6 7EA6 8C8B 8C8B 9D6B 4 24 89E8 8461 8461 7C62 3E98 EE06 8051 9AE3 6 1 25 7BD9 85E0 85E0 9D62 9D62 8CD2 8CD2 AC89 2 26 7DB4 81E8 81E8 97C8 97C8 97FB 97FB 9FE6 3 27 80A2 7EA6 7EA6 8C8B 8C8B 9D6B 9D6B 8E22 4 28 8461 7C62 3E98 EE06 8051 9AE3 9AE3 7B6D 7 1 29 85E0 9D62 9D62 8CD2 8CD2 AC89 AC89 9C5E 2 30 81E8 97C8 97C8 97FB 97FB 9FE6 9FE6 AA6B 3 31 7EA6 8C8B 8C8B 9D6B 9D6B 8E22 8E22 B0DB 4 32 3E98 EE06 8051 9AE3 9AE3 7B6D 7B6D AE2A

[0116] demodulation ROM2(conbined ROM for synchronization) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q r i 1 2 3 4 5 6 7 8 0 1 1 9D62 8CD2 8CD2 AC89 AC89 9C5E 9C5E 7AA0 2 2 97C8 97FB 97FB 9FE6 9FE6 AA6B AA6B 6DA9 3 3 8C8B 9D6B 9D6B 8E22 8E22 B0DB B0DB 650B 4 4 8051 9AE3 9AE3 7B6D 7B6D AE2A AE2A 6216 1 1 5 8CD2 AC89 AC89 9C5E 9C5E 7AA0 7AA0 AF18 2 6 97FB 9FE6 9FE6 AA6B AA6B 6DA9 6DA9 A655 3 7 9D6B 8E22 8E22 B0DR B0DB 650B 650B 9945 4 8 9AE3 7B6D 7B6D AE2A AE2A 6216 6216 89E8 2 1 9 AC89 9C5E 9C5E 7AA0 7AA0 AF18 AF18 7BD9 2 10 9FE6 AA6B AA6B 6DA9 6DA9 A655 A655 7DB4 3 11 8E22 B0DB B0DB 650B 650B 9945 9945 80A2 4 12 7B6D AE2A AE2A 6216 6216 89E8 89E8 8461 3 1 13 9C5E 7AA0 7AA0 AF18 AF18 7BD9 7BD9 85E0 2 14 AA6B 6DA9 6DA9 A655 A655 7DB4 7DB4 81E8 3 15 B0DB 650B 650B 9945 9945 80A2 80A2 7EA6 4 16 AE2A 6216 6216 89E8 89E8 8461 8461 7C62 4 1 17 7AA0 AF18 AF18 7BD9 7BD9 85E0 85E0 9D62 2 18 6DA9 A655 A655 7DB4 7DB4 81E8 81E8 97C8 3 19 650B 9945 9945 80A2 80A2 7EA6 7EA6 8C8B 4 20 6216 89E8 89E8 8461 8461 7C62 3E98 EE06 5 1 21 AF18 7BD9 7BD9 85E0 85E0 9D62 9D62 8CD2 2 22 A655 7DB4 7DB4 81E8 81E8 97C8 97C8 97FB 3 23 9945 80A2 80A2 7EA6 7EA6 8C8B 8C8B 9D6B 4 24 89E8 8461 8461 7C62 3E98 EE06 8051 9AE3 6 1 25 7BD9 85E0 85E0 9D62 9D62 8CD2 8CD2 AC89 2 26 7DB4 81E8 81E8 97C8 97C8 97FB 97FB 9FE6 3 27 80A2 7EA6 7EA6 8C8B 8C8B 9D6B 9D6B 8E22 4 28 8461 7C62 3E98 EE06 8051 9AE3 9AE3 7B6D 7 1 29 85E0 9D62 9D62 8CD2 8CD2 AC89 AC89 9C5E 2 30 81E8 97C8 97C8 97FB 97FB 9FE6 9FE6 AA6B 3 31 7EA6 8C8B 8C8B 9D6B 9D6B 8E22 8E22 B0DB 4 32 3E98 EE06 8051 9AE3 9AE3 7B6D 7B6D AE2A

[0117] demodulation ROM2-4(for synchronization) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 8051 9AE3 7B6D AE2A 6216 89E8 8461 7C62 1 9AE3 7B6D AE2A 6216 89E8 8461 7C62 8051 2 7B6D AE2A 6216 89E8 8461 7C62 8051 9AE3 3 AE2A 6216 89E8 8461 7C62 8051 9AE3 7B6D 4 6216 89E8 8461 7C62 8051 9AE3 7B6D AE2A 5 89E8 8461 7C62 8051 9AE3 7B6D AE2A 6216 6 8461 7C62 8051 9AE3 7B6D AE2A 6216 89E8 7 7C62 8051 9AE3 7B6D AE2A 6216 89E8 8461

[0118] these four demodulation ROM are conbined to one by the method descrived in this example demodulation ROM2-1(for synchronization, p = 0 block of ROM1 is arranged address incrementally) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 9D62 8CD2 AC89 9C5E 7AA0 AF18 7BD9 85E0 1 8CD2 AC89 9C5E 7AA0 AF18 7BD9 85E0 9D62 2 AC89 9C5E 7AA0 AF18 7BD9 85E0 9D62 8CD2 3 9C5E 7AA0 AF18 7BD9 85E0 9D62 8CD2 AC89 4 7AA0 AF18 7BD9 85E0 9D62 8CD2 AC89 9C5E 5 AF18 7BD9 85E0 9D62 8CD2 AC89 9C5E 7AA0 6 7BD9 85E0 9D62 8CD2 AC89 9C5E 7AA0 AF18 7 85E0 9D62 8CD2 AC89 9C5E 7AA0 AF18 7BD9

[0119] demodulation ROM2-2(for synchronization) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 97C8 97FB 9FE6 AA6B 6DA9 A655 7DB4 81E8 1 97FB 9FE6 AA6B 6DA9 A655 7DB4 81E8 97C8 2 9FE6 AA6B 6DA9 A655 7DB4 81E8 97C8 97FB 3 AA6B 6DA9 A655 7DB4 81E8 97C8 97FB 9FE6 4 6DA9 A655 7DB4 81E8 97C8 97FB 9FE6 AA6B 5 A655 7DB4 81E8 97C8 97FB 9FE6 AA6B 6DA9 6 7DB4 81E8 97C8 97FB 9FE6 AA6B 6DA9 A655 7 81E8 97C8 97FB 9FE6 AA6B 6DA9 A655 7DB4

[0120] demodulation ROM2-3(for synchronization) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 8C8B 9D6B 8E22 B0DB 650B 9945 80A2 7EA6 1 9D6B 8E22 B0DB 650B 9945 80A2 7EA6 8C8B 2 8E22 B0DB 650B 9945 80A2 7EA6 8C8B 9D6B 3 B0DB 650B 9945 80A2 7EA6 8C8B 9D6B 8E22 4 650B 9945 80A2 7EA6 8C8B 9D6B 8E22 B0DB 5 9945 80A2 7EA6 8C8B 9D6B 8E22 B0DB 650B 6 80A2 7EA6 8C8B 9D6B 8E22 B0DB 650B 9945 7 7EA6 8C8B 9D6B 8E22 B0DB 650B 9945 80A2

[0121] to combine these four demodulation ROM of which data placed at timing proper over-sampling poditic demodulation ROM1(convined data of 4 ROMs) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q r i 1 2 3 4 5 6 7 8 0 1 1 9D62 997B B890 AF21 B887 AF2C 9D5F 997E 2 2 97C8 8F11 AB0D 95FE AB0A 9606 97C7 8F13 3 3 8C8B 829F 8FED 77F6 8FF2 77F7 8C8D 829F 4 4 8051 794B 726B 6175 7274 616F 8055 7949 1 1 5 8CD2 960D 9094 A6DE 908D A6DF 8CD0 960D 2 6 97FB 9CEC AB83 B772 AB79 B77A 97F7 9CEF 3 7 9D6B 9C08 B8A5 B549 B89B B554 9D68 9C0B 4 8 9AE3 93BE B28B A147 B286 A151 9AE2 93C1 2 1 9 AC89 A7C5 C013 C67C BC7F A548 A8F2 8674 2 10 9FE6 89B7 99AC 903D 84E5 7B3E 8B0C 74A5 3 11 8E22 6BC3 6F9D 588F 4E89 54B8 6CF0 67E9 4 12 7B6D 54FB 4BD1 2C95 263E 3ACF 55BA 6341 3 1 13 9C5E C288 CE26 EE2C F38D DC95 C1E6 B0E2 2 14 AA6B C9D6 DF89 FECC FED7 DFA4 C9D5 AA92 3 15 B0DB C21D DCC6 F3E7 EE98 CE87 C2BD 9C9C 4 16 AE2A AD31 C685 D00F C6A7 AD48 AE4C 8A4B 4 1 17 7AA0 4A52 24CB 0B29 0A60 1643 3AAC 6533 2 18 6DA9 3CA4 14B6 0089 008B 14E7 3CD9 6DF2 3 19 650B 3AC7 1689 0AE5 0BB3 256C 4AD0 7AFF 4 20 6216 4505 29FE 28A9 2A23 454E 626F 8A5C 5 1 21 AFI8 BA97 C855 BF2F BD43 A4C7 944D 7AAC 2 22 A655 9E73 9CC5 8A81 886D 765D 7516 6DB3 3 23 9945 7F37 6E5F 55C5 53DB 4AF7 5916 6510 4 24 89E8 61A6 4436 2907 2791 2935 4491 6216 6 1 25 7BD9 6FD3 63CC 6B74 7B6F 8F46 9374 939A 2 26 7DB4 776C 7737 8187 923E A0B3 9CB0 968D 3 27 80A2 80A0 849C 9857 A85B B02F A455 9870 4 28 8461 8AA7 967B ADF0 BBE0 BC66 A9B9 9918 7 1 29 85E0 73E0 5AAE 429D 3DB5 4D55 68F1 7E95 2 30 81E8 6D32 5198 3CC8 3CB9 516F 6D11 81D2 3 31 7EA6 6902 4D5F 3D9B 4266 5A69 73B1 85C6 4 32 7C62 67AB 4E60 4503 4E3E 677E 7C40 8A1B

[0122] the circuit block diagram which use demodulation ROM1 conbined data in number of over-sampling represented in FIG. 5, and output the demodulation data which is accumulated and divided by α. demodulation ROM1-4(data of demodulation matrix4 exchanged to positive Hex. data) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 8051 794B 726B 6175 7274 616F 8055 7949 1 9AE3 93BE B28B A147 B286 A151 9AE2 93C1 2 7B6D 54FB 4BD1 2C95 263E 3ACF 55BA 6341 3 AE2A AD31 C685 D00F C6A7 AD48 AE4C 8A4B 4 6216 4505 29FE 28A9 2A23 454E 626F 8A5C 5 89E8 61A6 4436 2907 2791 2935 4491 6216 6 8461 8AA7 967B ADF0 BBE0 BC66 A9B9 9918 7 7C62 67AB 4E60 4503 4E3E 677E 7C40 8A1B

[0123] before the multiplication, data stored in ROM is exchanged to the number indicating positive or negat

[0124] also the modulation data is exchanged to the number having positive signe.

[0125] where Di is 8 bit modulation data, equation of exchange is

2×Di−255

[0126] when the result of calculation prosess is output, the data is done by inverse exchange. demodulation ROM1-1(data of demodulation matrix1 exchanged to positive Hex. data) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 9D62 997B B890 AF21 B887 AF2C 9D5F 997E 1 8CD2 960D 9094 A6DE 908D A6DF 8CD0 960D 2 AC89 A7C5 C013 C67C BC7F A548 A8F2 8674 3 9C5E C288 CE26 EE2C F38D DC95 C1E6 B0E2 4 7AA0 4A52 24CB 0B29 0A60 1643 3AAC 6533 5 AF18 BA97 C855 BF2F BD43 A4C7 944D 7AAC 6 7BD9 6FD3 63CC 6B74 7B6F 8F46 9374 939A 7 85E0 73E0 5AAE 429D 3DB5 4D55 68F1 7E95

[0127] demodulation ROM1-2(data of demodulation matrix2 exchanged to positive Hex. data) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 97C8 8F11 AB0D 95FE AB0A 9606 97C7 8F13 1 97FB 9CEC AB83 B772 AB79 B77A 97F7 9CEF 2 9FE6 89B7 99AC 903D 84E5 7B3E 8B0C 74A5 3 AA6B C9D6 DF89 FECC FED7 DFA4 C9D5 AA92 4 6DA9 3CA4 14B6 0089 008B 14E7 3CD9 6DF2 5 A655 9E73 9CC5 8A81 886D 765D 7516 6DB3 6 7DB4 776C 7737 8187 923E A0B3 9CB0 968D 7 81E8 6D32 5198 3CC8 3CB9 516F 6D11 81D2

[0128] demodulation ROM1-3(data of demodulation matrix3 exchanged to positive Hex. data) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 8C8B 829F 8FED 77F6 8FF2 77F7 8C8D 829F 1 9D6B 9C08 B8A5 B549 B89B B554 9D68 9C0B 2 8E22 6BC3 6F9D 588F 4E89 54B8 6CF0 67E9 3 B0DB C21D DCC6 F3E7 EE98 CE87 C2BD 9C9C 4 650B 3AC7 1689 0AE5 0BB3 256C 4AD0 7AFF 5 9945 7F37 6E5F 55C5 53DB 4AF7 5916 6510 6 80A2 80A0 849C 9857 A85B B02F A455 9870 7 7EA6 6902 4D5F 3D9B 4266 5A69 73B1 85C6

[0129] stored data in modulation ROM (data of modulation matrix exchanged to positive Hex. data) p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q r i 1 2 3 4 5 6 7 8 0 1 1 E584 CDF5 F0E5 BC50 F63D B105 FA7E A521 2 2 A107 FBA9 C728 EA65 DA73 DA90 EA75 C710 3 3 4EE0 F632 8CA1 FF5F B0DF F64D D144 E2E3 4 4 110C BFD2 4F1F F64C 7FD6 FFFF B116 F636 1 1 5 0121 6F0A 1D25 D14F 4ED4 F62D 8CAF FF5E 2 6 25B5 2546 027E 9923 2552 DA56 6731 FD92 3 7 6FA5 010D 0574 5B08 09A3 B0B8 43D6 F0FA 4 8 C05A 115A 2555 25A6 0000 7FAD 25A6 DAAA 2 1 9 F66D 4F70 5A9A 0596 09E2 4EAE 0F3C BC8F 2 10 FB80 A19E 98B1 0268 25C6 2535 0284 993F 3 11 CD79 E5E3 D0F6 1CDC 4F6D 0993 0096 73C3 4 12 7F63 FFFF F620 4EB4 807B 0000 099D 4F54 3 1 13 318E E524 FF6A 8C2E B177 09F2 1CD3 2F15 2 14 042E A070 EAA5 C6C8 DAE7 25E4 388F 15CA 3 15 0A0A 4E4F BCB5 F0AF F67B 4F93 5A70 05A3 4 16 40B5 10BE 8073 FFFF FFFF 80A4 7F8C 0000 4 1 17 9190 0136 4415 F11B F5FD B19D A4B2 0560 2 18 DB27 2624 15DB C788 D9FE DB04 C6B0 154A 3 19 FF06 7040 00AB 8D14 B046 F68B E29A 2E62 4 20 EE56 C0E1 0987 4F89 7F32 FFFE F609 4E7F 5 1 21 AFFE F6A8 2E57 1D6E 4E3C F5ED FF52 72DD 2 22 5DCA FB57 666B 0295 24DE D9E1 FDA8 985D 3 23 19BE CCFC A489 0553 0964 B020 F130 BBC4 4 24 0001 7EC7 DA07 2503 0001 7F08 DAFC DA07 6 1 25 1B3B 3112 FA47 5A2B 0A22 4E17 BCF5 F08C 2 26 6027 0406 FDAD 9840 263B 24C1 99B0 FD64 3 27 B240 0A47 E36C D09C 5005 0955 7435 FF74 4 28 EF8E 413D B1B5 F5F3 8120 0001 4FBF F68D 7 1 29 FEB3 922B 7444 FF75 B20E 0A32 2F6E E375 2 30 D96B DB95 3997 EAE4 DB5B 2659 160B C7CF 3 31 8F23 FF19 0F87 BD1B F6B9 502B 05C5 A5FD 4 32 3E98 EE06 0001 80E6 FFFD 8149 0001 80E6

[0130] and above data are delayed one over-sampling and demodulated using the demodulation matrix 1˜4 the mean value of demodulation data are q r j x_(j) 0 1 1 3.9847 2 2 7.2244 1 1 3 −2.1027 2 4 −10.1946 2 1 5 1.5695 2 6 11.9217 3 1 7 −1.3175 2 8 −13.1774

[0131] the out put adjusted data by which is mentioned adjustment circuit in this paper using above adjustment parameter q r j x_(j) 0 1 1 1.0945 2 2 14.9236 1 1 3 −0.679 2 4 −14.8599 2 1 5 1.0464 2 6 15.1295 3 1 7 −1.0984 2 8 −15.1392

[0132] above data is rounded as below q r j x_(j) 0 1 1 1 2 2 15 1 1 3 −1 2 4 −15 2 1 5 1 2 6 15 3 1 7 −1 2 8 −15

[0133] this demodulation data is same as the modulation data of transmission side.

[0134] above data are changed a little by the noise of line to demodulation circuit i d_(i) 1 3.4752 2 3.8076 3 0.2641 4 −6.2598 5 −13.2246 6 −18.1845 7 −17.6699 8 −12.0553 9 −3.1223 10 6.2118 11 11.5807 12 11.6888 13 5.5937 14 −2.9461 15 −10.6135 16 −12.4051 17 −6.4104 18 5.6919 19 21.8561 20 34.7286 21 40.1087 22 35.2151 23 19.0815 24 −3.6546 25 −26.993 26 −44.5621 27 −52.648 28 −49.4096 29 −36.3798 30 −18.107 31 0.7936 32 15.1676

[0135] modulation data are as x₁=x₅=x₆=15, x₃=x₇=−1, x₄=x₈=−15 in practical communication, modulaed data are i d_(i) 1 3.8853 2 0.4624 3 −6.1088 4 −13.2972 5 −17.9469 6 −17.698 7 −12.1194 8 −3.0384 9 6.1155 10 11.7103 11 11.4849 12 5.6252 13 −3.1445 14 −10.5474 15 −12.4635 16 −6.6923 17 5.9496 18 21.6735 19 34.9428 20 40.4428 21 35.062 22 19.1293 23 −3.5679 24 −26.9999 25 −44.8891 26 −52.7427 27 −49.1863 28 −36.1509 29 −17.9151 30 0.5586 31 15.3163 32 3.3394

[0136] and above data are delayed one over-sampling and demodulated using the demodulation matrix 1˜4 the mean value of demodulation data are q r j x_(j) 0 1 1 10.9686 2 2 4.0607 1 1 3 12.015 2 4 8.7172 2 1 5 12.6055 2 6 11.1354 3 1 7 13.4383 2 8 12.7259

[0137] the parameters of adjustment data are determined as p {overscore (D)}_(2p+1) ²(test) + {overscore (D)}_(2p+2) ²(test) 0 136.8002 1 220.3448 2 282.8956 3 342.5367 p {overscore (D)}_(2p+1)(test) + {overscore (D)}_(2p+2)(test) 0 15.0293 1 20.7322 2 23.7409 3 26.1642 p {overscore (D)}_(2p+1)(test) − {overscore (D)}_(2p+2)(test) 0 6.908 1 3.2977 2 1.4702 3 0.7124

[0138] above data are changed a little by the noise of line to demodulation circuit i d_(i) 1 79.7745 2 81.1090 3 65.0050 4 37.0460 5 5.6597 6 −21.4443 7 −37.4286 8 −41.5717 9 −36.3611 10 −26.2757 11 −17.1823 12 −11.3325 13 −10.1673 14 −10.4529 15 −9.9276 16 −5.3161 17 3.3021 18 12.8503 19 21.3957 20 23.7096 21 19.4529 22 10.3367 23 −1.1009 24 −9.2484 25 −10.4869 26 −3.8698 27 7.3148 28 18.5145 29 24.5774 30 21.4752 31 9.2918 32 −9.5909

[0139] modulated data for DA converter input i d_(i) 1 81.1867 2 65.2033 3 37.1970 4 5.5872 5 −21.2066 6 −37.4566 7 −41.6358 8 −36.2772 9 −26.3720 10 −17.0527 11 −11.5364 12 −10.1358 13 −10.6513 14 −9.8615 15 −5.3745 16 3.0202 17 13.1080 18 21.2131 19 23.9238 20 19.7870 21 10.1836 22 −1.0532 23 −9.1617 24 −10.4938 25 −4.1968 26 7.2201 27 18.7377 28 24.8063 29 21.6671 30 9.0569 31 −9.4422 32 79.6386

[0140] the frequency of sub-carriers is determined as below

[0141] r=1 α=4 ρf₀=2.5×10 and according to below equation ${\sum\limits_{q = 0}^{n - 1}\quad {\cos^{2}\frac{2\pi \quad f_{\rho}}{\rho \times \alpha \times f_{0}}}} = {\sum\limits_{q = 0}^{n - 1}\quad {\sin^{2}\frac{2\pi \quad f_{\rho}}{\rho \times \alpha \times f_{0}}}}$

[0142] ƒ₀=1.0423 MHz

[0143] ƒ₁=0.7809 MHz

[0144] ƒ₂=0.6255 MHz

[0145] ƒ₃=0.4684 MHz

[0146] therefore ρ=2.399 is got.

[0147] if the bit wide of the modulation data is only one bit, the trancemission speed is ${2.399 \times 4 \times 1.0423 \times \frac{1}{4}} = {2.5\quad {Mbps}}$

[0148] elements data of modulation matris and demodulation matrix 1˜4 should be changed to positive hex number to store in ROM, and the example method about cos θ is descrived as following $\frac{{\cos \quad \theta} + 1}{2} \times 65535$ $\frac{\begin{matrix} \begin{matrix} {\text{data~~of~~demodulati~~on~~matrix} +} \\ \text{(absolute~~maximum~~negative} \end{matrix} \\ \text{data~~of~~demodulati~~on~~matrix)} \end{matrix}}{\begin{matrix} \text{(absolute~~maximum~~negative} \\ \text{data~~of~~demodulati~~on~~matrix)} \end{matrix} \times 2} \times 65535$

[0149] by above equation, cos θ values of modulation matrix and of demodulation matris 1˜4 are changed to positive dezimal number and are changed to hex number and are stored in ROM.

[0150] each address of ROMS are i 1 and q and the number of port is j and 16 bits wide in this example.

[0151] at first in test communication, modulation is done as all modulation data are 15(F)

x₁=x₂=x₃= . . . x₈=15

[0152] demodulation matrix 2 p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 0.3351 0.1206 0.8091 0.2909 0.8088 0.2916 0.3351 0.1209 1 0.3403 0.4618 0.8213 1.1147 0.8203 1.1154 0.3400 0.4621 2 0.5350 −0.0107 0.3819 0.1500 −0.1295 −0.3669 0.0219 −0.5293 3 0.7938 1.5670 2.1010 2.8705 2.8715 2.1036 1.5669 0.7976 4 −0.7012 −1.9079 −2.8909 −3.3878 −3.3876 −2.8863 −1.9030 −0.6944 5 0.6936 0.5000 0.4588 0.0094 −0.0421 −0.4867 −0.5184 −0.7003 6 −0.3065 −0.4610 −0.5643 −0.2119 0.1997 0.5554 0.4565 0.3052 7 −0.2032 −0.7132 −1.3929 −1.9051 −1.9064 −1.3964 −0.7161 −0.2052

[0153] demodulation matrix 3 p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 0.0585 −0.1856 0.1414 −0.4483 0.1418 −0.4482 0.0586 −0.1856 1 0.4740 0.4398 1.1442 1.0612 1.1432 1.0622 0.4737 0.4401 2 0.0979 −0.7478 −0.6531 −1.2202 −1.4673 −1.3148 −0.7190 −0.8427 3 0.9522 1.3771 2.0331 2.6026 2.4718 1.6826 1.3924 0.4541 4 −0.9133 −1.9539 −2.8462 −3.1332 −3.1132 −2.4799 −1.5594 −0.3733 5 0.3722 −0.2687 −0.6832 −1.2887 −1.3362 −1.5549 −1.2076 −0.9129 6 −0.2344 −0.2343 −0.1359 0.3499 0.7442 0.9366 0.6446 0.3516 7 −0.2834 −0.8163 −1.4967 −1.8847 −1.7665 −1.1753 −0.5529 −0.1078

[0154] demodulation matrix4 p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 −0.2423 −0.4151 −0.5848 −1.0019 −0.5839 −1.0024 −0.2420 −0.4153 1 0.4116 0.2357 0.9936 0.5686 0.9931 0.5695 0.4114 0.2360 2 −0.3624 −1.3085 −1.5340 −2.3026 −2.4589 −1.9525 −1.2902 −0.9572 3 0.8860 0.8623 1.4855 1.7206 1.4888 0.8646 0.8893 0.0034 4 −0.9862 −1.7021 −2.3676 −2.4008 −2.3642 −4.6954 −0.9780 0.0047 5 −0.0059 −0.9965 −1.7211 −2.3903 −2.4266 −2.3861 −1.7128 −0.9862 6 −0.1421 0.0126 0.3041 0.8816 1.2246 1.2372 0.7773 0.3678 7 −0.3392 −0.8491 −1.4717 −1.7021 −1.4747 −0.8531 −0.3421 −0.0012

[0155] method of demodulation matrix creation

[0156] for example, matrix of which size is 8 columns is created by picking the line number r=1 p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 0.7931 0.6091 0.8820 0.4712 0.9238 0.3830 0.9570 0.2901 1 −0.9912 −0.1326 −0.7723 0.6353 −0.3841 0.9233 0.0990 0.9951 2 0.9253 −0.3791 −0.2922 −0.9564 −0.9228 −0.3852 −0.8811 0.4730 3 −0.6132 0.7900 0.9955 0.0951 0.3863 −0.9224 −0.7747 −0.6324 4 0.1377 −0.9905 −0.4680 0.8837 0.9219 0.3874 0.2870 −0.9579 5 0.3744 0.9273 −0.6380 −0.7700 −0.3885 0.9214 0.9948 −0.1022 6 −0.7868 −0.6172 0.9553 −0.2956 −0.9210 −0.3896 0.4758 0.8796 7 0.9898 0.1428 −0.0916 0.9958 0.3907 −0.9205 −0.6299 0.7767

[0157] the demodulation matrix is inverse matrix of the above matrix demodulation matrix 1 p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q 1 2 3 4 5 6 7 8 0 0.4731 0.3770 1.1419 0.9096 1.1411 0.9106 0.4728 0.3773 1 0.0658 0.2928 0.1586 0.7069 0.1580 0.7070 0.0656 0.2929 2 0.8459 0.7289 1.3268 1.4849 1.2389 0.6677 0.7577 −0.0911 3 0.4480 1.3872 1.6732 2.4612 2.5938 2.0283 1.3717 0.9529 4 −0.3821 −1.5709 −2.4948 −3.1259 −3.1454 −2.8527 −1.9564 −0.9097 5 0.9092 1.1926 1.5310 1.3061 1.2585 0.6558 0.2499 −0.3810 6 −0.3523 −0.6481 −0.9441 −0.7554 −0.3619 0.1264 0.2291 0.2326 7 −0.1056 −0.5488 −1.1693 −1.7617 −1.8824 −1.4975 −0.8178 −0.2849

[0158] modulation matrix p 0 1 2 3 r 1 2 1 2 1 2 1 2 j q r i 1 2 3 4 5 6 7 8 0 1 1 0.7931 0.6091 0.8820 0.4712 0.9238 0.3830 0.9570 0.2901 2 2 0.2580 0.9661 0.5559 0.8312 0.7067 0.7075 0.8317 0.5552 3 3 −0.3839 0.9234 0.0987 0.9951 0.3818 0.9242 0.6349 0.7726 4 4 −0.8669 0.4985 −0.3819 0.9242 −0.0012 1.0000 0.3834 0.9236 1 1 5 −0.9912 −0.1326 −0.7723 0.6353 −0.3841 0.9233 0.0990 0.9951 2 6 −0.7053 −0.7089 −0.9805 0.1964 −0.7084 0.7058 −0.1939 0.9810 3 7 −0.1276 −0.9918 −0.9574 −0.2888 −0.9247 0.3807 −0.4702 0.8826 4 8 0.5030 −0.8643 −0.7084 −0.7058 −1.0000 −0.0024 −0.7060 0.7082 2 1 9 0.9253 −0.3791 −0.2992 −0.9564 −0.9228 −0.3852 −0.8811 0.4730 2 10 0.9648 0.2629 0.1929 −0.9812 −0.7050 −0.7092 −0.9804 0.1970 3 11 0.6050 0.7962 0.6325 −0.7746 −0.3796 −0.9251 −0.9954 −0.0958 4 12 −0.0051 1.0000 0.9228 −0.3852 0.0036 −1.0000 −0.9248 −0.3805 3 1 13 −0.6132 0.7900 0.9955 0.0951 0.3863 −0.9224 −0.7747 −0.6324 2 14 −0.9675 0.2530 0.8332 0.5530 0.7101 −0.7041 −0.5579 −0.8299 3 15 −0.9214 −0.3886 0.4744 0.8803 0.9256 −0.3785 −0.2931 −0.9561 4 16 −0.4941 −0.8694 0.0036 1.0000 1.0000 0.0048 −0.0032 −1.0000 4 1 17 0.1377 −0.9905 −0.4680 0.8837 0.9219 0.3874 0.2870 −0.9579 2 18 0.7125 −0.7017 −0.8292 0.5589 0.7033 0.7109 0.5526 −0.8335 3 19 0.9925 −0.1225 −0.9948 0.1022 0.3774 0.9261 0.7706 −0.6373 4 20 0.8617 0.5074 −0.9256 −0.3785 −0.0060 1.0000 0.9223 −0.3864 5 1 21 0.3744 0.9273 −0.6380 −0.7700 −0.3885 0.9214 0.9948 −0.1022 2 22 −0.2679 0.9635 −0.1999 −0.9798 −0.7118 0.7024 0.9816 0.1908 3 23 −0.7993 0.6009 0.2854 −0.9584 −0.9265 0.3763 0.8841 0.4673 4 24 −0.9999 −0.0102 0.7033 −0.7109 −1.0000 −0.0072 0.7105 0.7037 6 1 25 −0.7868 −0.6172 0.9553 −0.2956 −0.9210 −0.3896 0.4758 0.8796 2 26 −0.2481 −0.9687 0.9819 0.1894 −0.7016 −0.7126 0.2002 0.9798 3 27 0.3933 −0.9194 0.7768 0.6297 −0.3752 −0.9270 −0.0927 0.9957 4 28 0.8719 −0.4896 0.3885 0.9215 0.0084 −1.0000 −0.3775 0.9260 7 1 29 0.9898 0.1428 −0.0916 0.9958 0.3907 −0.9205 −0.6299 0.7767 2 30 0.6980 0.7161 −0.5500 0.8352 0.7135 −0.7007 −0.8281 0.5605 3 31 0.1174 0.9931 −0.8786 0.4775 0.9274 −0.3740 −0.9551 0.2962 4 32 −0.5118 0.8591 −1.0000 0.0072 1.0000 0.0096 −1.0000 0.0064

EXAMPLE

[0159] [Effect of Invention]

[0160] The effect of this invention applied to DSL of metal twist pair is described below. The parameter of the modulation and demodulation system is differ from example and determined below. Number of carrier frequency n = 16 Number of over-sampling α = 8 Bit wide of modulation data A = 8

Numbers of basic sampling in one wave form ρ Most high frequency of sub-carrier ƒ₀

[0161] defined as ρƒ₀=12.5 MHz. Sampling frequency CLK of DA and AD converter is

CLK=ρ×α×ƒ₀=12.5×8=100 MHz

[0162] Transmission speed is ${{CLK} \times \frac{A}{\alpha}} = {{100 \times \frac{8}{8}} = {100\quad {Mbps}}}$

[0163] About frequency of sub-carrier concern in the frequency range of 6.0 MHz˜0.09 MHz ${\sum\limits_{q = 0}^{31}{\cos^{2}\frac{2\quad \pi \quad f_{p}}{8 \times 12.5}\left( {{8q} + 1} \right)}} \cong {\sum\limits_{q = 0}^{31}{\sin^{2}\frac{2\quad \pi \quad f_{p}}{8 \times 12.5}\left( {{8q} + 1} \right)}}$

[0164] The frequency is determined as $\begin{matrix} {f_{0} = {0.0950\quad {MHz}}} & {f_{1} = {0.473\quad {MHz}}} & {f_{2} = {0.852\quad {MHz}}} & {f_{3} = {1.231\quad {MHz}}} \\ {f_{4} = {1.610\quad {MHz}}} & {f_{5} = {1.989\quad {MHz}}} & {f_{6} = {2.368\quad {MHz}}} & {f_{7} = {2.747\quad {MHz}}} \\ {f_{8} = {3.314\quad {MHz}}} & {f_{9} = {3.693\quad {MHz}}} & {f_{10} = {4.072\quad {MHz}}} & {f_{11} = {4.451\quad {MHz}}} \\ {f_{12} = {4.830\quad {MHz}}} & {f_{13} = {5.208\quad {MHz}}} & {f_{14} = {5.588\quad {MHz}}} & {f_{15} = {5.966\quad {MHz}}} \end{matrix}$

SIMPLE PROVE OF FIG.

[0165] [FIG. 1] Modulation and demodulation total system block diagram.

[0166] [FIG. 2] Block diagram of modulation.

[0167] [FIG. 3] Block diagram of demodulation.

[0168] [FIG. 4] Block diagram of phase and magnitude adjustment.

[0169] [FIG. 5] Block diagram of distributed time phase and magnitude adjustment.

[0170] [FIG. 6] Block diagram of demodulation for synchronization.

[0171] [FIG. 7] Block diagram of adjustment signal.

[0172] [FIG. 8] Block diagram of adjustment signal.

[0173] [FIG. 9] Block diagram of adjustment signal.

SIMPLE PROVE OF SYMBOL

[0174] DFF data flip-flop.

[0175] CLK clock.

[0176] ROM read only memory.

[0177] TFF toggle flip-flop. 

1. The numbers of sub-carrier frequency are n, and by using proper positive integer α, modulation circuit is constructed, by 2n numbers of modulation ROM of which address are 2αn, by 2n numbers of multiplier of 2n number of modulation data and data of modulation ROM as product, by accumulator summing together all data from multipliers, and by DA converter converting the data to analog from accumulator. Demodulation circuit is constructed, by ad converter of which sampling frequency is same as da converter, by 2n numbers of ROM1 and 4n numbers of ROM2 of which address is 2αn, by 2n numbers of multipliers by which the data from AD converter is multiplied with the data of each ROM1, by 2n numbers of accumulator by which the products of multiplier are accumulated and reset at every 2αn data, by 4n numbers of multiplier by which the data from AD converter is multiplied with the data of each ROM2, by 4αn numbers of accumulator by which the products of the multiplier are accumulated partially by α number and reset at every 2αn data, and by the phase adjustment circuit using the accumulated data of cosine and sine wave of same sub-carrier frequency. 2n numbers of modulation Rom store the columns element data independently of modulation matrix of which elements are the value of trigonometric sine and cosine of sub-carrier frequencies in the address as sampling order, 2n number of demodulation ROM1, of which lines and columns are 2αn and 2n, store the element data of combined matrix in α number position of inverse matrices which are made from the α numbers matrices of 2n lines and 2n columns picking up the line of modulation ROM according a number. 4n numbers of demodulation ROM2 store one pair of ROM1 belonging to sine and cosine of same sub-carrier frequency and store columns data which are shifted α number respectively to the end of this pair data of ROM1. 4αn numbers of accumulated partially data by α number are adjusted about phase and are sent in accordance with shifting number to comparator and the address counter for demodulation is reset as the top of same data series near to a meets the data of the top of minimum shift number. Such system is claimed.
 2. The modulation and demodulation system included in claim 1, of which sub-carrier frequency is determined so as to become minimum difference of the accumulated square value of the cosine data and the sine data picked up from modulation ROM by a interval of same sub-carrier frequency.
 3. The modulation and demodulation system included in claim 1, of which synchronization circuit is different as following, of which every two series of modulation data are same and in specified trigonometric function in a sub-carrier for synchronization this is different from next series, of which demodulation ROM for synchronization is picked up from demodulation ROM1 responding synchronization sub-carrier for modulation and specified the address by continuous number being different from demodulation ROM address, of which data of demodulation ROM for synchronization is multiplied with AD converted data and accumulated for one round address number from every address as starting and send amount value at every time, and which have comparator comparing the amount value and next one find out difference for the synchronization signal.
 4. The modulation and demodulation system included in claim 1, which have the same demodulation circuits described in claim 1 and a first detection circuit constructed by partial oscillator, mixer and mid-frequency filter, and which have such modulation block that output the same signal as DA converter output described in claim 1 at the output of mid-frequency filter by proper modulation method.
 5. The modulation and demodulation system include in claim 1, of which element of the modulation ROM are the products of the element of trigonometric modulation ROM and proper element of same size matrix at same position of each matrix, and of which element of demodulation ROM is inverted matrix of this above modulation ROM.
 6. The modulation and demodulation system included in claim 1, of which sub-carrier frequency is determined so as to become minimum difference between the accumulated value of the cosine data to the power m and the accumulated value of the sine data to the power m of which cosine and sine are picked up from modulation ROM by a interval of same sub-carrier frequency, by positive integer number of m. 